**Risk ratios**, **rate ratios** and **odds ratios** are 3 different measures of association between an exposure and a disease or other health outcome^{1-3}. Risk ratios and rate ratios are derived from randomized trials or cohort studies while odds ratios are calculated from case-control studies.

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#### Risk ratios

Risk ratios are the simplest of these 3 measures. A risk is the probability of a particular outcome occurring during a specific length of time and is calculated by dividing the number of people in a group who develop the outcome by the total number of people in the group:

risk = number of people with the outcome / total number in the group

For example, a cohort study may be done to compare how often people ≥65 years of age die after contracting a new flu strain compared to those under 65 during flu season. The “exposure” in this example is age ≥65 years old. If 900 of 1,000 people ≥65 years of age died, then the risk or probability of death in older people who contracted the new strain is 900/1,000=0.90 or 90%. If 100 of 1,000 people under 65 died, then the risk or probability of death in people under 65 who contracted the new strain is 100/1,000=0.10 or 10%.

The risk ratio compares the likelihood of an outcome between the exposed and unexposed and is calculated using the formula:

risk ratio = risk in exposed / risk in unexposed

The risk ratio in the hypothetical flu example above is: risk in exposed / risk in unexposed = 0.90/0.10 = 9. This means that people infected with the new flu strain who are ≥ 65 years old have a 9 times greater chance of dying than those infected who are less than 65 years old.

Exposure may be protective: a randomized trial may be done to compare how often vaccinated and unvaccinated subjects become ill with flu during flu season. If 100 of 1,000 participants in the vaccinated group become infected, then the risk or probability of disease in the vaccinated is 0.10 or 10%. If 900 of 1,000 subjects in the unvaccinated group become infected, then the risk or probability of disease in the unvaccinated is 0.90 or 90%. The risk ratio in this hypothetical example is: risk in exposed / risk in unexposed = 0.10/0.90 = 1/9. This means that the probability of a vaccinated person becoming ill is 1/9th^{ }the probability of an unvaccinated person becoming ill.

Risk ratios are appropriate when the time for the development of an outcome is short, but when a disease takes many years to develop, patients may be lost to follow-up or die of other diseases. If we followed our theoretical 2,000 vaccinated and unvaccinated subjects for 30 years instead of 3 months, some participants would move to another location and become difficult to contact while others would die. When this happens, the number of study participants changes over time, making the denominator for our risk calculation unstable: therefore, risk cannot be calculated.

#### Rate ratios

Rate ratios are used instead of risk ratios when the number of subjects in the cohort changes over time. Instead of counting the number of individuals in the cohort, the time that each subject spends as an at-risk study participant is counted. For example, in a 30-year trial

a participant that remains healthy for 30 years contributes 30 person-years

a participant that develops the disease or outcome of interest after 20 years contributes 20 person-years

a participant that dies of another disease after 20 years contributes 20 person-years

a participant that is lost to follow-up after 10 years contributes 10 person-years

The incidence rate is analogous to risk and is a measure of how often people develop disease per unit of time. The numerator for risk and incidence rate is the same: the number of study participants who develop the disease or outcome of interest. In calculating risk, the denominator is the total number of study participants, but for incidence rate (IR), the denominator is the total number of person-years:

incidence rate = number of participants developing disease / total amount of time at risk

The rate ratio is analogous to the risk ratio and is calculated using the formula:

rate ratio = incidence rate in the exposed / incidence rate in the unexposed

In 1956, Doll and Hill published a cohort study in the British Medical Journal comparing the death rates of smokers and non-smokers from lung cancer after 53 months of follow-up in male physicians ≥35 years of age^{4}. Among smokers, (the exposed), there were 83 lung cancer deaths during 98,090 person-years, for an incidence rate of 83/98,090 or 0.84 lung cancer deaths/1,000 person-years. Among the non-smokers (the unexposed), there was 1 lung cancer death during 15,107 person-years, for an incidence rate of 1/15,107 or 0.07 lung cancer deaths/1,000 person-years. The rate ratio for the “Doctor’s Study” as published in 1956 is IR exposed/IR unexposed = 0.84/0.07 = 12. This means that smokers died of lung cancer at 12 times the rate of non-smokers.

Like risk ratios, rate ratios can also be used to evaluate protective exposures. In the National Lung Screening Trial (NLST), there were 356 lung cancer deaths and 144,103 person-years in subjects screened with low-dose CT and 443 lung cancer deaths and 143,368 person-years in subjects screened with chest X-ray^{5}. The death rate due to lung cancer was 247 per 100,000 person-years in the CT group and 309 deaths per 100,000 person-years in the X-ray group. The risk ratio was IR exposed/IR unexposed = 247/309 = 0.80. This means that during the NLST, those screened with CT died of lung cancer at 80% of the rate of those screened with radiography^{5}.

#### Odds ratios

Odds ratios are the most complex of these 3 measures of association. Odds themselves are ratios, so an odds ratio is a ratio of 2 ratios. Odds are derived from probabilities using the formula:

odds = probability of an event occurring / probability of an event not occurring

or

odds = probability of an event occurring / (1-probability of an event occurring)

In words, odds are the probability of some event occurring divided by the probability of that event not occurring. Using our flu strain example above, the odds of people ≥65 years of age dying of flu = probability of dying / probability of not dying = 0.90/0.10 = 9. The odds of people <65 years of age dying of flu = probability of dying / probability of not dying = 0.10/0.90 = 1/9.

The odds ratio compares the odds of exposure between cases and controls and is calculated using the formula:

odds ratio = odds of exposure in cases / odds of exposure in controls

Odds ratios are often calculated from the so-called cross product of cells from a 2x2 table where cell “a” is the number of exposed cases, cell “b” is the number of unexposed cases, “c” is the number of exposed controls and “d” is the number of unexposed controls:

odds ratio = (ad) / (bc)

Returning to the new flu strain example, the odds ratio for dying of flu in subjects ≥65 years of age compared to those <65 years is (0.90/0.10) / (0.10/0.90) = 81. In other words, the odds of being ≥65 years old are 81 times greater than the odds of being under 65 years of age in those dying of the flu.

Like risk and rate ratios, odds ratios can also be calculated for protective exposures and like these other ratios, an odds ratio <1 indicates a protective exposure.

The above odds ratio calculations are meant to illustrate the mechanics of calculating an odds ratio and to explain the meaning of an odds ratio: normally, we wouldn't calculate odds and odds ratios from cohort study data.

In 1950, Doll and Hill published a case-control study in the British Medical Journal comparing the exposure to smoking in subjects with lung cancer to hospitalized controls without lung cancer^{6}. There were 688 exposed cases (cell “a”), 21 unexposed controls (cell “b”), 650 exposed controls (cell “c”), and 59 unexposed controls (cell ”d”), resulting in an odds ratio = (ad) / (bc) = (688x59) / (650x21) = 40,592/13,650 = 2.97. In words, the odds of smoking among the subjects with lung cancer (cases) were 2.97 times greater than the odds of smoking among the subjects without lung cancer (controls). This study was one of the earliest that showed an association between smoking and lung cancer.

#### The Rare Disease Assumption

An odds ratio will approximate the risk ratio if the rare disease assumption is met. Using the 2x2 table format discussed above, the risk ratio (RR=risk in exposed/risk in unexposed) can be expressed as the following ratio:

RR=a/(a+c)**/**b/(b+d)

If “a” and “b” are both small numbers, then the equation for RR can be approximated by RR= a/c**/**b/d which rearranges to RR=(ad)/(bc) which is the "cross product" formula for the odds ratio. The rare disease assumption is often made when disease prevalence is less than 10%

#### Practical points

risk ratios, rate ratios and odds ratios >1 indicate a harmful exposure

risk ratios, rate ratios and odds ratios =1 indicate that a health outcome is equally likely among the exposed and unexposed

risk ratios, rate ratios and odds ratios <1 indicate a protective exposure

if a disease is rare, the risk ratio and the odds ratio are nearly equal